^ (circumflex)
The exponentiation operator.
Syntax

a^b

a
is a number or a matrix 
b
is a number


A^n

A
is a set or a list 
n
is a small positive integer

Description
Real numbers
If a
is a real (or complex) number and n
a positive integer, a^n
is equal to the product of n
factors each equal to a
. This also applies for n = 0
unless a = 0
as well; 0^0
is undefined and will generate an error (“zero raised to the power of zero”).
If n
is a negative integer, a^n
is defined as 1/a^−n
unless a = 0
which is again undefined and will generate a program error (“division by zero”).
If a
is a nonnegative real number and q
a positive integer, a^(1/q)
is defined as the unique nonnegative real number ξ
such that ξ^q = a
.
Finally, if a
is a nonnegative real number, p
is an integer and q
a positive integer, a^(p/q)
is defined as [a^(1/q)]^p
, subject to the above restrictions if a = 0
.
This defines a^b
completely for floatingpoint numbers.
For a > 0
, a^b
equals exp(b⋅ln(a))
.
Complex numbers
If a
and b
are complex numbers with a ≠ 0
, then a^b
is defined as exp(b⋅ln(a))
. Notice that this also defines a^b
for a
and b
real, a < 0
, and b
nonintegral.
If a = 0
and Re(b) > 0
, then a^b = 0
.
Matrices
If A
is a square matrix and n
a nonnegative integer, then A^n
is the product of n
factors all equal to A
, the empty product being the identity matrix the size of A
. This also applies if A
is the zero matrix.
If n
is a negative integer, then A^n = inv(A)^n = inv(A^n)
assuming A
is nonsingular. If A
is singular, A^n
is undefined and generates an error.
A^b
is not defined for nonintegral b
. However, if A
is a positive semidefinite matrix, its unique positive semidefinite square root can be computed as √A
or sqrt(A)
. Computing A^(1/2)
is not supported.
Cartesian products
If A
is a set or a list and n
a small positive integer, then A^n
returns the Cartesian product A × ⋯ × A
, the product containing exactly n
factors.
No hard limit on the size of n
is enforced, but since the cardinality of A^n
equals #A^n
, you will run out of memory if #A
is not very small and n
not a very small integer. For instance, if #A = 1000
and n = 4
, #A^n = 1 000 000 000 000
which is far more than the system can handle.
Notes
The ^
operator is implemented by the power
function.
Examples
∑((−1)^n/n!, n, 0, 100)
0.367879441171 (=e⁻¹)
π^i
0.413292116102 + 0.910598499213⋅i
A ≔ ❨❨3, i❩, ❨−i, 2❩❩
⎛ 3 i⎞ ⎝−i 2⎠
A^5
⎛ 450 275⋅i⎞ ⎝−275⋅i 175⎠
A^−2
⎛ 0.2 −0.2⋅i⎞ ⎝ 0.2⋅i 0.4⎠
ScatterPlot([0, 10, 1]^3)